WEBVTT 00:00:01.110 --> 00:00:03.310 What is a function? 00:00:03.970 --> 00:00:08.470 Now, one definition of a function is that a function is a 00:00:08.470 --> 00:00:12.970 rule that Maps 1 unique number to another unique number. So In 00:00:12.970 --> 00:00:17.845 other words, If I start off with a number and I apply my 00:00:17.845 --> 00:00:21.595 function, I finish up with another unique number. So for 00:00:21.595 --> 00:00:25.345 example, let's suppose that my function added three to any 00:00:25.345 --> 00:00:30.595 number I could start off with a #2. I apply my function and I 00:00:30.595 --> 00:00:32.845 finish up with the number 5. 00:00:33.440 --> 00:00:37.318 Start off with the number 8 and I apply my function and I finish 00:00:37.318 --> 00:00:38.703 up with the number 11. 00:00:39.370 --> 00:00:45.730 And if I start off with a number X and I apply my function, I 00:00:45.730 --> 00:00:50.818 finish up with the number X +3 and we can show this 00:00:50.818 --> 00:00:56.330 mathematically by writing F of X equals X plus three, where X is 00:00:56.330 --> 00:01:00.570 our inputs, which we often called the arguments of the 00:01:00.570 --> 00:01:06.506 function and X +3 is our output. Now suppose I had an argument of 00:01:06.506 --> 00:01:09.898 two, I could write down F of two 00:01:09.898 --> 00:01:14.220 equals. 2 + 3, which gives us an output of five. 00:01:15.080 --> 00:01:21.112 Suppose I had an argument of eight. I could write down F of 00:01:21.112 --> 00:01:26.680 eight equals 8 + 3, which gives me an output of 11. 00:01:27.330 --> 00:01:33.322 And suppose I had an argument of minus six. I could write F of 00:01:33.322 --> 00:01:39.804 minus 6. Equals minus 6 + 3, which would give me minus three 00:01:39.804 --> 00:01:41.136 for my output. 00:01:42.030 --> 00:01:48.810 And if I had an argument of zed, I could write down F of said 00:01:48.810 --> 00:01:50.618 equals zed plus 3. 00:01:51.600 --> 00:01:57.264 And likewise, if I had an argument of X squared, I could 00:01:57.264 --> 00:02:01.984 write down F of X squared equals X squared +3. 00:02:02.720 --> 00:02:07.062 Now it is with me pointing out here that's a lower first sight. 00:02:07.062 --> 00:02:10.736 It might appear that we can choose any value for argument. 00:02:10.736 --> 00:02:14.744 That's not always the case, as we shall see later, but because 00:02:14.744 --> 00:02:19.086 we do have some choice on what number we can pick the argument 00:02:19.086 --> 00:02:22.092 to be, this is sometimes called the independent variable. 00:02:22.710 --> 00:02:26.910 And because output depends on our choice of arguments, the 00:02:26.910 --> 00:02:29.010 output is sometimes called the 00:02:29.010 --> 00:02:35.650 dependent variable. Now let's have a look at an example 00:02:35.650 --> 00:02:39.430 F of X equals 3 X 00:02:39.430 --> 00:02:42.510 squared. Minus 4. 00:02:43.080 --> 00:02:48.274 Now, as we said before, X is our input which we call the argument 00:02:48.274 --> 00:02:50.129 and that is the independent 00:02:50.129 --> 00:02:54.791 variable. And our output which is 3 X squared minus four is 00:02:54.791 --> 00:02:55.814 our dependent variable. 00:02:56.860 --> 00:03:01.720 Now we can choose different values for arguments, which is 00:03:01.720 --> 00:03:08.524 often a good place to start when we get function like this. So F 00:03:08.524 --> 00:03:15.814 of zero will give us 3 * 0 squared takeaway 4, which is 0 - 00:03:15.814 --> 00:03:18.730 4, which is just minus 4. 00:03:19.540 --> 00:03:21.927 If we start off with an argument 00:03:21.927 --> 00:03:28.820 of one. We get F of one equals 3 * 1 squared takeaway 4, 00:03:28.820 --> 00:03:34.045 which is just three takeaway four which gives us minus one. 00:03:34.310 --> 00:03:41.240 If we have an argument of two F of two equals 3 * 2 squared 00:03:41.240 --> 00:03:46.784 takeaway, four switches 3 * 4, which is 12 takeaway. Four gives 00:03:46.784 --> 00:03:52.264 us 8. And as I said before, there's no reason why we can't 00:03:52.264 --> 00:03:56.736 include and negative arguments. So if I put F of minus one in. 00:03:56.750 --> 00:04:03.122 The three times minus one squared takeaway 4, which is 3 * 00:04:03.122 --> 00:04:08.432 1 gives us three takeaway. Four gives us minus one. 00:04:08.450 --> 00:04:11.950 And F of minus 2. 00:04:12.010 --> 00:04:17.639 Would give us three times minus 2 squared takeaway 4 which is 3 00:04:17.639 --> 00:04:23.268 * 4 gives us 12 and take away four will give us 8. 00:04:24.120 --> 00:04:28.436 So what are we going to do with these results? Well, one thing 00:04:28.436 --> 00:04:33.416 we can do is put them into a table to help us plot the graph 00:04:33.416 --> 00:04:37.732 of the function. So if we do a table of X&F of X. 00:04:38.560 --> 00:04:43.455 The values were chosen for RX column, which is. Our arguments 00:04:43.455 --> 00:04:47.015 are minus 1 - 2 zero one and 00:04:47.015 --> 00:04:54.697 two. So minus 2 - 1 zero 12. 00:04:55.430 --> 00:05:00.374 And the corresponding outputs are 8 - 1. 00:05:01.500 --> 00:05:05.180 Minus 4 - 1 and 00:05:05.180 --> 00:05:09.624 8. And we can use this as I said to help us plot the 00:05:09.624 --> 00:05:10.576 graph of the function. 00:05:11.680 --> 00:05:18.316 So just to copy the table down again. So we've got handy. 00:05:18.990 --> 00:05:25.750 We have minus 2 - 1 zero 12. 00:05:26.420 --> 00:05:32.468 8 - 1 - 4 - 1 and 8. 00:05:33.140 --> 00:05:35.360 So we plot our graph. 00:05:36.240 --> 00:05:42.540 Of F of X on the vertical axis, so output and arguments. 00:05:43.690 --> 00:05:47.058 On the horizontal axis. 00:05:47.060 --> 00:05:50.108 So we've got 00:05:50.108 --> 00:05:54.240 one too. Minus 1 - 2. 00:05:55.250 --> 00:05:57.749 And we've got. 00:05:57.750 --> 00:06:00.830 Minus 1 - 2 - 3 - 4. 00:06:01.820 --> 00:06:04.298 I'm going off. 00:06:05.560 --> 00:06:12.640 Up to 81234567 and eight. So our first point is minus 2 00:06:12.640 --> 00:06:16.180 eight so we can put the 00:06:16.180 --> 00:06:22.318 appear. Our second point, minus 1 - 1 should appear down here. 00:06:23.480 --> 00:06:26.459 0 - 4. 00:06:26.460 --> 00:06:29.220 Here one and minus one. 00:06:30.480 --> 00:06:34.752 Yeah, but up on two and eight which will appear over here 00:06:34.752 --> 00:06:39.024 and we can see we can draw a smooth curve through these 00:06:39.024 --> 00:06:42.228 points, which will be the graph of the function. 00:06:49.090 --> 00:06:53.398 OK, now why are we drawing a graph of a function? Because 00:06:53.398 --> 00:06:58.065 this is quite useful to us because we can now read off the 00:06:58.065 --> 00:07:02.014 output of a function for any given arguments straight off the 00:07:02.014 --> 00:07:04.527 graph without the need to do any 00:07:04.527 --> 00:07:08.025 calculations. So for example, if we look at two and arguments of 00:07:08.025 --> 00:07:11.175 two, we know that's going to give us 8 before I do work that 00:07:11.175 --> 00:07:15.570 out. But if we looked and we wanted to figure out. 00:07:16.360 --> 00:07:21.148 But the output would be when the argument was one point 5. If we 00:07:21.148 --> 00:07:25.594 follow our lineup and across you can see that that gives us a 00:07:25.594 --> 00:07:29.698 value between 2:00 and 3:00 for the output, and if we substitute 00:07:29.698 --> 00:07:33.460 in 1.5 back into our original expression for the function, you 00:07:33.460 --> 00:07:36.880 can see actually gives us an exact value of 2.75. 00:07:37.820 --> 00:07:42.008 Now earlier on when I discussed uniqueness, I said that a unique 00:07:42.008 --> 00:07:46.894 inputs had to give us a unique output and by that what we mean 00:07:46.894 --> 00:07:51.082 is that for any given argument we should get only one output. 00:07:51.670 --> 00:07:55.044 One of the benefits of having a graph of a function is that we 00:07:55.044 --> 00:07:56.490 can check this using our ruler. 00:07:57.460 --> 00:08:00.916 If we line our ruler up vertically and we move it left 00:08:00.916 --> 00:08:02.356 and right across the graph. 00:08:03.350 --> 00:08:07.074 We can make sure that the rule I only have across is the graph 00:08:07.074 --> 00:08:08.138 wants at any point. 00:08:09.050 --> 00:08:13.329 And as we can see, that's clearly the case in this 00:08:13.329 --> 00:08:17.608 example. And when that happens, the graph is a valid function. 00:08:19.300 --> 00:08:22.978 Now, if we had the example. 00:08:23.530 --> 00:08:25.609 F of X. 00:08:26.310 --> 00:08:28.968 Equals root X. 00:08:30.040 --> 00:08:36.449 A good place to start is always to substitute in some values for 00:08:36.449 --> 00:08:39.407 the arguments, so F of 0. 00:08:39.420 --> 00:08:43.061 This gives us the square root of 0, which is 0. 00:08:43.740 --> 00:08:48.459 Half of one's own arguments of one will give us plus or minus 00:08:48.459 --> 00:08:50.274 one for the square root. 00:08:51.010 --> 00:08:53.410 F of two. 00:08:53.990 --> 00:09:01.070 Will give us plus or minus 1.4 just to one decimal place 00:09:01.070 --> 00:09:04.780 there. Half of 3. 00:09:05.320 --> 00:09:10.948 Which gives us the square root of 3 gives us plus or minus 1.7. 00:09:11.700 --> 00:09:18.190 An F4. Will give us square root of 4 which is just 00:09:18.190 --> 00:09:19.670 plus or minus 2. 00:09:20.240 --> 00:09:25.486 Now. If we try to put in any negative 00:09:25.486 --> 00:09:28.356 arguments here you can see that we're going to run 00:09:28.356 --> 00:09:30.939 into trouble because we have to try and calculate 00:09:30.939 --> 00:09:33.522 the square root of a negative number and we'll 00:09:33.522 --> 00:09:36.679 come back to this problem in a second. But for now, 00:09:36.679 --> 00:09:39.262 let's plot the points that we've got so far. 00:09:40.520 --> 00:09:45.668 So if we take out F of X axis vertical again. 00:09:45.670 --> 00:09:48.585 And our arguments access X 00:09:48.585 --> 00:09:54.416 horizontal. We've got 00:09:54.416 --> 00:09:56.729 1234. 00:10:00.630 --> 00:10:02.172 And there are vertical axis we 00:10:02.172 --> 00:10:04.940 have minus one. Minus 2. 00:10:06.200 --> 00:10:13.618 Plus one. Plus two points. We've got zero and zero. 00:10:14.870 --> 00:10:16.298 One on plus one. 00:10:17.230 --> 00:10:19.750 And also 1A minus one. 00:10:21.280 --> 00:10:24.504 We've got two and 00:10:24.504 --> 00:10:31.716 positive 1.4. So round about that and also to negative 00:10:31.716 --> 00:10:35.150 1.4. We've got three and 00:10:35.150 --> 00:10:42.645 positive 1.7. And we've got three and negative 1.7. 00:10:42.740 --> 00:10:45.428 And finally we have four and +2. 00:10:46.660 --> 00:10:48.950 And four and negative 2. 00:10:49.630 --> 00:10:54.700 OK, and we've got enough points here that we can draw a smooth 00:10:54.700 --> 00:10:56.260 curve through these points. 00:10:59.560 --> 00:11:01.260 At something it looks like. 00:11:02.510 --> 00:11:03.250 This. 00:11:05.200 --> 00:11:11.152 OK. Now. As usual, we will apply our 00:11:11.152 --> 00:11:15.663 ruler test to make sure that the function is valid and you can 00:11:15.663 --> 00:11:19.827 see straight away that when we line up all the vertically and 00:11:19.827 --> 00:11:22.950 move it across for any given positive arguments, we're 00:11:22.950 --> 00:11:26.073 getting two outputs. So obviously we need to do 00:11:26.073 --> 00:11:28.155 something about this to make the 00:11:28.155 --> 00:11:32.867 function valid. One way to get around this problem is by 00:11:32.867 --> 00:11:36.397 defining route X to take only positive values or 0. 00:11:37.230 --> 00:11:38.860 This is sometimes called the 00:11:38.860 --> 00:11:42.606 positive square root. So in effect we lose the bottom 00:11:42.606 --> 00:11:44.146 negative half of this graph. 00:11:44.750 --> 00:11:48.226 And obviously we also have the issue of the negative arguments, 00:11:48.226 --> 00:11:51.702 and since we can't take the square root of a negative 00:11:51.702 --> 00:11:55.810 number, we also have to exclude these from the X axis. Now when 00:11:55.810 --> 00:11:58.654 we start talking about these kind of restrictions, it's 00:11:58.654 --> 00:12:01.814 important that we use the right kind of mathematical language. 00:12:02.420 --> 00:12:07.110 So the set of possible inputs is what we call the domain, and the 00:12:07.110 --> 00:12:10.460 set of possible outputs is what we call the range. 00:12:11.120 --> 00:12:16.356 So in this case, when we've got RF of X equals the square root 00:12:16.356 --> 00:12:22.592 of X. We need to restrict our domain to be X is more than or 00:12:22.592 --> 00:12:26.706 equal to 0, 'cause we only wanted the positive values and 00:12:26.706 --> 00:12:31.568 zero. But we also notice that now because we've got rid of the 00:12:31.568 --> 00:12:33.438 bottom half of the graph. 00:12:34.340 --> 00:12:39.228 The only part of the range which are included are also more than 00:12:39.228 --> 00:12:44.868 or equal to 0. So range is defined by F of X more than or 00:12:44.868 --> 00:12:47.876 equal to 0. So now we have a 00:12:47.876 --> 00:12:53.253 valid function. So what will do now is just look at a couple 00:12:53.253 --> 00:12:56.683 more examples to pull together everything that we've done so 00:12:56.683 --> 00:12:58.741 far and will start with this 00:12:58.741 --> 00:13:06.128 one. Let's look at the function F of X equals 2 X squared 00:13:06.128 --> 00:13:08.224 minus three X +5. 00:13:08.770 --> 00:13:12.970 No, as usual, a good place to start when you get a function is 00:13:12.970 --> 00:13:14.770 to substitute in some values for 00:13:14.770 --> 00:13:19.730 the arguments. So let's start with that. So now arguments of 0 00:13:19.730 --> 00:13:24.110 would give us F of 0, which is 2 * 0 squared. 00:13:24.680 --> 00:13:31.480 Minus 3 * 0 + 5 which is just zero 00:13:31.480 --> 00:13:34.200 takeaway 0 + 5. 00:13:34.210 --> 00:13:38.760 So we get now pose A5 that if we had an argument of one. 00:13:39.470 --> 00:13:46.810 We get 2 * 1 squared takeaway 3 * 1 00:13:46.810 --> 00:13:53.000 + 5. Which gives us 2 * 1 here, which is 2. 00:13:53.880 --> 00:14:00.090 Take away 3 * 1 which is take away three and plus five. So two 00:14:00.090 --> 00:14:04.230 takeaway three is minus 1 + 5 gives us 4. 00:14:05.120 --> 00:14:08.464 OK, we look at an argument of two. 00:14:09.530 --> 00:14:13.118 We get 2 * 2 squared. 00:14:13.730 --> 00:14:15.880 Take away 3 * 2. 00:14:16.380 --> 00:14:23.911 I'm plus five which gives us 2 * 4, which is 8 and take away 3 * 00:14:23.911 --> 00:14:30.556 2 which is take away 6 and then forget our plus five at the end. 00:14:30.556 --> 00:14:33.214 So eight takeaway six is 2. 00:14:33.810 --> 00:14:36.310 +5 gives us 7. 00:14:36.900 --> 00:14:40.399 OK. If we look at an argument of 00:14:40.399 --> 00:14:46.084 three. Half of three gives us 2 * 3 squared. 00:14:47.420 --> 00:14:49.290 Take away 3 * 3. 00:14:49.810 --> 00:14:51.300 And +5. 00:14:52.470 --> 00:14:56.026 So this is 2 * 9 here 00:14:56.026 --> 00:15:01.604 18. Take away 3 * 3 which is take away nine and +5. 00:15:02.310 --> 00:15:07.205 So 18 takeaway 9 is 9 + 5 gives us 14. 00:15:08.120 --> 00:15:12.905 And last but not least, we can also include a negative 00:15:12.905 --> 00:15:18.125 arguments, so we'll put negative arguments of minus one. So F of 00:15:18.125 --> 00:15:22.040 minus one gives us two times minus 1 squared. 00:15:22.690 --> 00:15:26.050 Take away three times minus one. 00:15:27.370 --> 00:15:30.220 And of course, our +5. 00:15:30.790 --> 00:15:34.790 So two times minus one squared just gives us 2. 00:15:36.160 --> 00:15:39.706 Takeaway minus sorry takeaway three times negative one which 00:15:39.706 --> 00:15:42.070 just gives us a plus 3. 00:15:42.950 --> 00:15:45.128 And then we've got a +5. 00:15:45.930 --> 00:15:49.188 2 + 3 + 5 just gives us 10. 00:15:49.790 --> 00:15:53.390 And what we can do is as before, just put this into a table to 00:15:53.390 --> 00:15:55.310 make it nice and easy to make a 00:15:55.310 --> 00:16:01.595 graph of the function. So we put it into a table of X. 00:16:02.160 --> 00:16:03.940 And F of X. 00:16:04.680 --> 00:16:12.078 For arguments we had minus 10123. 00:16:12.940 --> 00:16:16.780 For Outputs, we had 10 five. 00:16:17.380 --> 00:16:20.605 4, Seven and 00:16:20.605 --> 00:16:24.380 14. OK, so let's see the graph of this function then. 00:16:26.530 --> 00:16:27.740 You start off with our. 00:16:28.430 --> 00:16:31.446 F of X on the vertical axis as 00:16:31.446 --> 00:16:33.820 before. An argument. 00:16:34.330 --> 00:16:36.540 X on the horizontal axis. 00:16:37.680 --> 00:16:41.104 And we've got over 2 - 1, The 00:16:41.104 --> 00:16:47.986 One. 2. And three, and on the vertical axis. 00:16:49.820 --> 00:16:52.748 Go to 15. 00:16:53.540 --> 00:16:59.616 OK, so our first point to plot is minus one and 10 which will 00:16:59.616 --> 00:17:02.220 give us something there zero and 00:17:02.220 --> 00:17:05.810 five. There's a point here. One 00:17:05.810 --> 00:17:08.532 and four. It gives the points 00:17:08.532 --> 00:17:10.389 here. Two and Seven. 00:17:11.150 --> 00:17:15.099 Should be around here and three and 14 which will be 00:17:15.099 --> 00:17:19.407 about here so we can see the kind of shape that this 00:17:19.407 --> 00:17:23.715 function is starting to take here. And we can draw in the 00:17:23.715 --> 00:17:24.074 graph. 00:17:34.700 --> 00:17:38.899 What we want is to say that every input gives us only one 00:17:38.899 --> 00:17:42.775 single output, so we can get our ruler again and just quickly 00:17:42.775 --> 00:17:47.297 check by going along and we can see that as we go along. Our 00:17:47.297 --> 00:17:49.235 rule across is the curve once 00:17:49.235 --> 00:17:53.388 and once only. Which means that this function is valid. 00:17:53.940 --> 00:17:59.044 However, an interesting point to note is this point here. The 00:17:59.044 --> 00:18:03.684 minimum point which actually occurs when X is North .75. 00:18:04.580 --> 00:18:11.184 So with X value of North .75 are outputs can take a minimum 00:18:11.184 --> 00:18:12.708 value of 3.875. 00:18:13.320 --> 00:18:17.254 So this means when we look at our domain and range, we need to 00:18:17.254 --> 00:18:20.064 make no restrictions on the domain because our function was 00:18:20.064 --> 00:18:27.040 valid. However. Our range has a minimum of 3.875, so we write 00:18:27.040 --> 00:18:30.510 this. As F of X equals. 00:18:31.290 --> 00:18:34.797 Two X squared minus three X +5. 00:18:36.040 --> 00:18:42.982 And the range F of X has always been more than or equal 00:18:42.982 --> 00:18:48.970 to 3.875. So for the next example. 00:18:50.020 --> 00:18:54.376 What would happen if we had a function F defined by? 00:18:54.960 --> 00:18:59.160 F of X equals one over X. 00:19:00.370 --> 00:19:04.495 Well, that's always the first stage is to substitute in some 00:19:04.495 --> 00:19:05.995 values for the arguments. 00:19:06.920 --> 00:19:09.040 So for F of one. 00:19:09.870 --> 00:19:13.687 The argument is one is 1 / 1 just gives US1. 00:19:14.980 --> 00:19:16.450 For Port F of two. 00:19:17.030 --> 00:19:18.730 We just get one half. 00:19:19.760 --> 00:19:22.358 F of three gives us 1/3. 00:19:22.910 --> 00:19:29.498 And F4 will give us 1/4. 00:19:30.520 --> 00:19:35.030 And as before, we can also look at some negative arguments. 00:19:35.710 --> 00:19:37.735 So if I look at F of minus one. 00:19:38.480 --> 00:19:42.528 Skip 1 divided by minus one, which is just minus one. 00:19:43.350 --> 00:19:45.190 F of minus 2. 00:19:45.740 --> 00:19:50.396 Is 1 divided by minus two, which just gives us minus 1/2? 00:19:51.880 --> 00:19:56.401 F of minus three. Same thing will give us minus 1/3. 00:19:57.080 --> 00:19:59.380 And F of minus 4. 00:19:59.940 --> 00:20:02.200 Will give us minus 1/4? 00:20:02.770 --> 00:20:05.714 Now if we look at F of 0. 00:20:05.960 --> 00:20:09.896 We have 1 / 00:20:09.896 --> 00:20:13.270 0. Which is obviously a problem 00:20:13.270 --> 00:20:17.208 for us. Because of this problem, we have to restrict 00:20:17.208 --> 00:20:21.720 our domain so that it does not include the arguments X equals 00:20:21.720 --> 00:20:26.232 0. So let's have a look at what the graph of this 00:20:26.232 --> 00:20:27.736 function actually looks like. 00:20:29.230 --> 00:20:33.169 So let's be 4F of X and are vertical axis for the Outputs. 00:20:34.240 --> 00:20:38.174 An argument sax on the horizontal axis. 00:20:39.640 --> 00:20:46.115 OK, so we've gone over here 1234. 00:20:47.790 --> 00:20:54.698 And minus 1 - 2 - 3 - 4 over here. 00:20:54.810 --> 00:20:56.430 As we go. 00:20:57.030 --> 00:20:58.990 All the web to one down to minus 00:20:58.990 --> 00:21:05.260 one. Has it as well? So we've got one and one. 00:21:06.390 --> 00:21:09.456 We've got 2 00:21:09.456 --> 00:21:12.510 1/2. 3 00:21:12.510 --> 00:21:16.046 1/3. 4 00:21:16.046 --> 00:21:23.556 one quarter. We've got minus 1 - 1. 00:21:23.650 --> 00:21:27.046 Minus 2 - 00:21:27.046 --> 00:21:33.750 1/2. Minus 3 - 1/3. 00:21:33.750 --> 00:21:37.218 A minus four and minus 1/4. 00:21:37.770 --> 00:21:42.114 OK, and obviously we've excluded 0 from our domain. As we said 00:21:42.114 --> 00:21:44.286 before. So if we join these 00:21:44.286 --> 00:21:47.030 points up. And a smooth curve. 00:21:48.830 --> 00:21:53.288 Get something that looks like this. 00:21:54.910 --> 00:21:59.618 Now, obviously we've excluded X equals 0 from our domain, but 00:21:59.618 --> 00:22:03.898 it's also worth noticing here. Thought there's nothing at the 00:22:03.898 --> 00:22:09.890 output of F of X equals 0, so that also ends up being excluded 00:22:09.890 --> 00:22:11.174 from the range. 00:22:11.790 --> 00:22:16.548 So we actually end up with F of X equals one over X. 00:22:17.450 --> 00:22:21.212 And we've got X not equal to 0 from the domain. 00:22:21.870 --> 00:22:26.690 And also. In the range F of X never equals 0 either. 00:22:28.910 --> 00:22:30.595 But what's actually happening at 00:22:30.595 --> 00:22:35.408 this point? X equals 0 when the arguments is zero. What is going 00:22:35.408 --> 00:22:40.000 on? Well, let's have a look and will start off by having a look 00:22:40.000 --> 00:22:43.280 what happens as we get closer and closer to 0. 00:22:43.820 --> 00:22:45.910 Now. 00:22:47.040 --> 00:22:52.352 If we start off with a value of 1/2 of one and remember F of X 00:22:52.352 --> 00:22:54.676 was just equal to one over X. 00:22:55.340 --> 00:23:01.085 Half of 1 just gives US1, so if I get closer to 0 again, let's 00:23:01.085 --> 00:23:03.000 look at half of 1/2. 00:23:03.000 --> 00:23:05.628 At 1 / 1/2. 00:23:06.200 --> 00:23:11.294 This one over 1/2 which just gives us 2. 00:23:11.300 --> 00:23:15.292 So about F of 00:23:15.292 --> 00:23:21.168 110th. It just gives us 1 / 1 tenth, which gives us 10. 00:23:21.820 --> 00:23:29.512 Half of one over 1000 will just give us 1 / 1 00:23:29.512 --> 00:23:33.314 over 1000. Which gives 00:23:33.314 --> 00:23:40.236 us 1000. What about one over 1,000,000, so F of 00:23:40.236 --> 00:23:42.564 one over a million? 00:23:42.570 --> 00:23:47.550 It's actually just in the same way as before, just going to 00:23:47.550 --> 00:23:49.210 give us a million. 00:23:49.220 --> 00:23:52.068 So we can see. 00:23:52.710 --> 00:23:57.316 The US we get closer and closer to zero from the right hand side 00:23:57.316 --> 00:23:59.619 as we saw on our graph before. 00:23:59.620 --> 00:24:03.230 We're getting closer and closer to positive Infinity to the 00:24:03.230 --> 00:24:05.035 graph goes off to positive 00:24:05.035 --> 00:24:09.135 Infinity that. What happens when we approach zero from the left 00:24:09.135 --> 00:24:10.965 hand side? Well, let's have a 00:24:10.965 --> 00:24:17.692 look. This is minus one, just gives us 1 divided by minus one, 00:24:17.692 --> 00:24:19.556 which is minus one. 00:24:20.500 --> 00:24:23.988 F of minus 1/2. 00:24:23.990 --> 00:24:27.790 It's going to be just one divided by minus 1/2. 00:24:28.330 --> 00:24:30.110 Just give his minus 2. 00:24:31.290 --> 00:24:34.401 Half of minus 00:24:34.401 --> 00:24:40.840 110th. Is 1 divided by minus 110th? 00:24:40.910 --> 00:24:47.150 It just gives us minus 10 and we can kind of see a pattern here. 00:24:47.150 --> 00:24:50.478 OK so F of minus one over 1000. 00:24:50.510 --> 00:24:52.760 Will actually give us minus 00:24:52.760 --> 00:24:59.384 1000. And F of minus one over 1,000,000. 00:24:59.390 --> 00:25:05.542 Actually gives us minus 00:25:05.542 --> 00:25:11.467 1,000,000. So you can see that as we approach zero from the 00:25:11.467 --> 00:25:12.895 left or outputs approaches 00:25:12.895 --> 00:25:17.606 negative Infinity. And as we approach zero from the right 00:25:17.606 --> 00:25:21.876 hand side are output approaches positive Infinity, and these are 00:25:21.876 --> 00:25:25.292 very different things. OK, for the last example. 00:25:25.940 --> 00:25:30.241 I just like to look at the function F defined by. 00:25:30.250 --> 00:25:37.576 F of X equals one over X minus two all squared. 00:25:38.310 --> 00:25:41.770 So as always with the examples we've done, it's worthwhile 00:25:41.770 --> 00:25:45.230 started off by looking at some different values for the 00:25:45.230 --> 00:25:51.234 arguments. So we start off with an argument of minus two, so 00:25:51.234 --> 00:25:56.876 half of minus two gives us one over minus 2 - 2 squared. 00:25:57.420 --> 00:26:03.803 Which gives us one over minus 4 squared, which is one over 16. 00:26:03.900 --> 00:26:07.388 F of minus one. 00:26:07.390 --> 00:26:14.628 Will give us one over minus 1 - 2 all squared, which gives us 00:26:14.628 --> 00:26:19.798 one over minus 3 squared which is one over 9. 00:26:19.820 --> 00:26:26.450 Now, FO arguments of zero will give us one over 0 - 2 00:26:26.450 --> 00:26:32.570 all squared, which is one over minus 2 squared, which works out 00:26:32.570 --> 00:26:34.100 as one quarter. 00:26:34.960 --> 00:26:41.932 And F of one will give us one over 1 - 2 00:26:41.932 --> 00:26:46.880 squared. Which is just one over minus one squared, which gives 00:26:46.880 --> 00:26:48.008 us just one. 00:26:48.580 --> 00:26:55.990 OK. Half of two gives us one over. 00:26:56.690 --> 00:27:01.730 2 - 2 or squared, which gives us one over 0 which presents us 00:27:01.730 --> 00:27:06.050 with exactly the same problem we had in the previous example when 00:27:06.050 --> 00:27:11.810 we had one over X and so we have to exclude X equals 2 from the 00:27:11.810 --> 00:27:19.370 domain. Half of three gives us one over 3 - 2 all squared. 00:27:20.220 --> 00:27:25.225 Which is one over 1 squared is just gives US1 again. 00:27:25.230 --> 00:27:32.440 After four gives us one over 4 - 2 or squared, which gives us 00:27:32.440 --> 00:27:33.985 one over 4. 00:27:34.110 --> 00:27:42.080 After 5. Gives us one over 5 - 2 or 00:27:42.080 --> 00:27:48.515 squared which gives us one over 3 squared which is 1 ninth and 00:27:48.515 --> 00:27:55.940 finally F of six gives us one over 6 - 2 all squared which is 00:27:55.940 --> 00:28:01.385 one over 4 squared which works out as one over 16. 00:28:01.940 --> 00:28:06.659 Now if we want to plot the graph of this function will probably 00:28:06.659 --> 00:28:09.200 need to put this into a table 00:28:09.200 --> 00:28:15.520 first. So as usual, do our table of X&F of X. 00:28:16.370 --> 00:28:23.104 OK, and we went from minus 2 - 1 zero all the way. 00:28:23.880 --> 00:28:26.298 Up to and arguments of sex. 00:28:27.160 --> 00:28:28.896 And the values we got for the 00:28:28.896 --> 00:28:31.932 Outputs. One over 00:28:31.932 --> 00:28:38.160 16. One 9th, one quarter and 00:28:38.160 --> 00:28:44.292 1:01. One quarter, 1 ninth and 00:28:44.292 --> 00:28:47.358 one over 16. 00:28:48.270 --> 00:28:51.550 So we plot that onto. 00:28:52.100 --> 00:28:54.120 The graph as before. 00:28:57.060 --> 00:29:00.606 So we have arguments going along the horizontal axis. 00:29:01.290 --> 00:29:03.600 And Outputs going along the vertical axis. 00:29:04.720 --> 00:29:12.112 We've gone from minus 1 - 2 over 00:29:12.112 --> 00:29:15.808 there 123456 along this 00:29:15.808 --> 00:29:18.810 way. And then going off, we've 00:29:18.810 --> 00:29:24.960 gone too. One up here, so put in a few of the marks 1/2. 00:29:25.540 --> 00:29:27.250 It's put in 1/4 that. 00:29:28.020 --> 00:29:34.728 Put in 3/4. OK, so we've got minus two and 116th, which is 00:29:34.728 --> 00:29:37.824 going to come in down here. 00:29:38.340 --> 00:29:40.580 Minus one and one 9th. 00:29:41.380 --> 00:29:47.108 For coming over here zero and one quarter. 00:29:47.110 --> 00:29:49.589 Over here. 1 on one. 00:29:50.930 --> 00:29:52.410 Right, the way up here? 00:29:53.200 --> 00:29:57.232 To an. Obviously this was the divide by zero, so we couldn't 00:29:57.232 --> 00:30:00.592 do anything with that. We've excluded, uh, from our domain. 00:30:01.690 --> 00:30:03.040 Three and one. 00:30:03.620 --> 00:30:07.280 Pay up. Four and one quarter. 00:30:08.840 --> 00:30:16.628 I'm here. Five and one, 9th and six and 116th. 00:30:16.630 --> 00:30:21.643 Because we've excluded X equals 2 from our domain. 00:30:23.370 --> 00:30:26.410 Put dotted line there, so that's an asymptotes. 00:30:27.820 --> 00:30:29.626 And we can draw our curve. 00:30:31.970 --> 00:30:33.298 Up through the points. 00:30:33.980 --> 00:30:36.770 On this side. 00:30:38.190 --> 00:30:42.493 And we can see differently to the other example where F of X 00:30:42.493 --> 00:30:44.479 is one over X this time. 00:30:45.090 --> 00:30:49.965 As we get approach to from both the left and from the right, 00:30:49.965 --> 00:30:54.090 both of the outputs are heading towards positive Infinity, so a 00:30:54.090 --> 00:30:57.840 little bit different, and also because we've excluded X equals 00:30:57.840 --> 00:31:00.465 2 from the domain of function is 00:31:00.465 --> 00:31:05.470 now valid. But most also notice that our range is never zero, 00:31:05.470 --> 00:31:07.315 and it's also never negative. 00:31:07.820 --> 00:31:10.826 So to write this out properly. 00:31:10.830 --> 00:31:16.394 Our function F of X equals one over X minus two all squared. 00:31:17.710 --> 00:31:18.829 And we said. 00:31:19.340 --> 00:31:20.804 They are domain is restricted so 00:31:20.804 --> 00:31:26.506 it doesn't include two. And our range is always more than 0. 00:31:27.640 --> 00:31:31.324 OK, so let's just recap on what we've done in this unit. 00:31:32.310 --> 00:31:34.452 So firstly, the definition of a 00:31:34.452 --> 00:31:39.480 function. And that was that. A function is a rule that Maps are 00:31:39.480 --> 00:31:42.880 unique number X to another unique number F of X. 00:31:44.230 --> 00:31:48.050 Secondly, was the idea that an argument is exactly the 00:31:48.050 --> 00:31:49.578 same as an input. 00:31:51.630 --> 00:31:55.950 Thirdly, we looked at the idea of independent and dependent 00:31:55.950 --> 00:32:00.443 variables. And we said that the input axe was the 00:32:00.443 --> 00:32:03.539 independent variable and the output was the dependent 00:32:03.539 --> 00:32:03.926 variable. 00:32:05.330 --> 00:32:10.230 4th, we looked at the domain and we said that the domain was the 00:32:10.230 --> 00:32:11.630 set of possible inputs. 00:32:12.620 --> 00:32:16.546 And finally we looked at the range and we said that the range 00:32:16.546 --> 00:32:18.358 was the set of possible outputs. 00:32:19.610 --> 00:32:21.514 So now you know how to define a 00:32:21.514 --> 00:32:24.997 function. And how to find the outputs of a function 00:32:24.997 --> 00:32:26.209 for a given argument?